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Τετραδική Ομάδα Klein
Τετραδική Ομάδα Klein Klein four-group, Vierergruppe, Rotations and reflections in two dimensions thumb|300px| [[Τετραδική Ομάδα Klein ]] thumb|300px| [[Τετραδική Ομάδα Klein ]] thumb|300px| [[Τετραδική Ομάδα Klein Αναστροφή ]] thumb|300px| [[Τετραδική Ομάδα Klein ---- The most intuitive way to understand the Klein four-group is through performing four simple rotation operations. First find a book and hold it in your hands with the title facing you, while imagining the book is sitting in cartesian coordinate space. Second, rotate the book 180 degrees about the x-axis, the horizontal line through the center of the book. Third, rotate the book 180 degrees about the y-axis, the vertical like through the center of the book. Finally, rotate the book 180 degrees about the z-axid, the line through the center perpendicular of the book. '' ]] thumb|300px| [[Τετραδική Ομάδα Klein ]] thumb|300px| [[Τετραδική Ομάδα Klein ]] thumb|300px| [[Τετραδική Ομάδα Klein ]] thumb|300px| [[Τετραδική Ομάδα Klein ---- Μεταπεριστροφή (screw rotation) Μεταντιστροφή (glide reflaxion = Transreflaxion) ]] thumb|300px| [[Ομαδοθεωρία ---- Αλγεβρική Ομάδα Γενική Γραμμική Ομάδα Ορθογώνια Ομάδα Μοναδιακή Ομάδα ---- Μαθηματική Αναπαράσταση Μαθηματική Μήτρα ]] - Μία Αλγεβρική Ομάδα. Ετυμολογία Η ονομασία ''"ομάδα" σχετίζεται ετυμολογικά με την λέξη "ομού". Εισαγωγή In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal and vertical reflection and 180-degree rotation), as the group of bitwise exclusive or operations on two-bit binary values, or more abstractly as Z2 × Z2, the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe (meaning four-group) by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter V or as K4. The Klein four-group, with four elements, is the smallest group that is not a cyclic group. There is only one other group of order four, up to isomorphism, the cyclic group of order 4. Both are abelian groups. The smallest non-abelian group is the symmetric group of degree 3, which has order 6. Ανάλυση The Klein group's Cayley table is given by: The Klein four-group is also defined by the group presentation : \mathrm{V} = \langle a,b \mid a^2 = b^2 = (ab)^2 = e \rangle. All non-identity elements of the Klein group have order 2, thus any two non-identity elements can serve as generators in the above presentation. The Klein four-group is the smallest non-cyclic group. It is however an abelian group, and isomorphic to the dihedral group of order (cardinality) 4, i.e. D4 (or D2, using the geometric convention); other than the group of order 2, it is the only dihedral group that is abelian. The Klein four-group is also isomorphic to the direct sum , so that it can be represented as the pairs under component-wise addition modulo 2 (or equivalently the bit strings under bitwise XOR); with (0,0) being the group's identity element. The Klein four-group is thus an example of an elementary abelian 2-group, which is also called a Boolean group. The Klein four-group is thus also the group generated by the symmetric difference as the binary operation on the subsets of a powerset of a set with two elements, i.e. over a field of sets with four elements, e.g. \{\emptyset, \{\alpha\}, \{\beta\}, \{\alpha, \beta\}\} ; the empty set is the group's identity element in this case. Another numerical construction of the Klein four-group is the set with the operation being multiplication modulo 8. Here a'' is 3, ''b is 5, and is . The Klein four-group has a representation as 2x2 real matrices with the operation being matrix multiplication: : e =\begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}\,\, a =\begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}\,\, b =\begin{pmatrix} -1&0\\ 0&1 \end{pmatrix}\,\, c =\begin{pmatrix} -1&0\\ 0&-1 \end{pmatrix} Υπάρχουν µε προσέγγιση ισοµορφίας δύο µόνον οµάδες τάξης 4: * η κυκλική τάξης 4 και * η οµάδα Klein. Υποσημειώσεις Εσωτερική Αρθρογραφία * Ομάδα * Γράφημα Cayley * Ομαδοθεωρία * Ορθογώνια Ομάδα * Μοναδιακή Ομάδα * Ετερωτική Ομάδα * Αναπαράσταση * Μήτρα * Τοπολογική Συνομολογία Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *thatsmaths.com *Understanding Symmetries and Geometry through Dance, L. Olliverrie Κατηγορία:Αλγεβρικές Ομάδες